<p>Matrices of order \(2 \times 2\) are formed by using the elements of the set \(A = \{-2, -1, 0, 1, 2\}\), then probability that matrix is either symmetric or skew-symmetric, is greater than:</p>
<p>(a) \(\dfrac{1}{10}\)</p>
<p>(b) \(\dfrac{2}{10}\)</p>
<p>(c) \(\dfrac{3}{10}\)</p>
<p>(d) \(\dfrac{4}{10}\)</p>
Step-by-Step Solution
Key Concept: A 2×2 matrix is symmetric if a₁₂ = a₂₁, and skew-symmetric if a₁₂ = -a₂₁. For skew-symmetric matrices, diagonal elements must be 0. These conditions are mutually exclusive, so count favorable cases separately and add them.
<p><strong>Step 1: Count total matrices</strong></p><p>A 2×2 matrix has 4 independent entries. Each can be chosen from 5 elements of set A = {-2, -1, 0, 1, 2}.</p><p>Total matrices = 5⁴ = 625</p><p><strong>Step 2: Count symmetric matrices</strong></p><p>For a symmetric matrix, a₁₂ = a₂₁. Independent choices: a₁₁, a₁₂, a₂₂ (3 positions)</p><p>Symmetric matrices = 5³ = 125</p><p><strong>Step 3: Count skew-symmetric matrices</strong></p><p>For skew-symmetric: a₁₂ = -a₂₁ and a₁₁ = a₂₂ = 0</p><p>Choices: a₁₁ = 0 (1 way), a₂₂ = 0 (1 way), a₁₂ ∈ A (5 ways), then a₂₁ = -a₁₂</p><p>Only if -a₁₂ ∈ A: From {-2, -1, 0, 1, 2}, for each a₁₂, -a₁₂ is also in A</p><p>Skew-symmetric matrices = 1 × 1 × 5 = 5</p><p><strong>Step 4: Apply inclusion-exclusion</strong></p><p>A matrix cannot be both symmetric and skew-symmetric (only zero matrix, but 0 appears on diagonal)</p><p>Favorable outcomes = 125 + 5 = 130</p><p><strong>Step 5: Calculate probability</strong></p><p>P = 130/625 = 26/125 = 0.208</p><p>∴ Probability is greater than 0.2 or 1/5 (Answer: B)</p>
Correct Answer: B